Value-Based Distance Between Information Structures

We define the distance between two information structures as the largest possible difference in value across all zero-sum games. We provide a tractable characterization of distance and use it to discuss the relation between the value of information in games versus single-agent problems, the value of additional information, informational substitutes, complements, or joint information. The convergence to a countable information structure under value-based distance is equivalent to the weak convergence of belief hierarchies, implying, among other things, that for zero-sum games, approximate knowledge is equivalent to common knowledge. At the same time, the space of information structures under the value-based distance is large: there exists a sequence of information structures where players acquire increasingly more information, and $\epsilon$ > 0 such that any two elements of the sequence have distance of at least $\epsilon$. This result answers by the negative the second (and last unsolved) of the three problems posed by J.F. Mertens in his paper Repeated Games , ICM 1986.